On the interaction of stationary crossflow vortices and Tollmien-Schlichting waves in the boundary layer on a rotating disc

Abstract

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete among themselves. Here we consider the interaction of two types of instability mode (at an asymptotically large Reynolds number) which can occur in the flow above a rotating disc. In particular, we exam in the interaction between lower-branch Tollmien-Schlichting (TS) waves and the upper-branch, stationary, inviscid crossflow vortex, whose asymptotic structure has been described. This problem is studied in the context of investigating the effect of the vortex on the stability characteristics of a small TS wave. Essentially, it is found that the primary effect is felt through the modification to the mean flow induced by the presence of the vortex. Initially, the TS wave is taken to be linear in character and we show (for the cases of both a stationary vortex with a viscous-type (linear) critical layer structure and one with a nonlinear critical layer) that the vortex can exhibit both stabilizing and destabilizing effects on the TS wave and the nature of this influence is wholly dependent upon the orientation of this latter instability. Further, we examine the problem with a larger TS wave, whose size is chosen so as to ensure that this mode is nonlinear in its own right. An amplitude equation for the evolution of the TS wave is derived which admits solutions corresponding to finite amplitude, stable, travelling waves.